Problem 21 - Calculus: One and Several Variables

Question

Write an equation for the line with slope 5 and \(y\) -intercept 2

Step-by-Step Solution

Answer

The equation of the line with a slope 5 and a y-intercept 2 is \(y = 5x + 2\).

Step 1: Identify the Slope and Y-Intercept

From the problem, it's given that the slope \(m = 5\) and the y-intercept \(b = 2\)

Step 2: Substitute into Slope-Intercept Equation

Substitute the given slope and y-intercept into the slope-intercept form equation \(y = mx + b\). Thus we get \(y = 5x + 2\)

Key Concepts

Slope-Intercept Form

The slope-intercept form is a specific way of writing the equation of a line on a graph. This form is particularly useful because it clearly shows both the slope and the y-intercept of the line, making it easy to graph and understand. It is written as: \[ y = mx + b \]In this equation, \( m \) represents the slope, and \( b \) represents the y-intercept. Knowing this format allows you to quickly determine how steep the line is (the slope) and where it crosses the y-axis (the y-intercept). When given a slope and y-intercept, plug these values directly into the equation to obtain the line equation quickly. This simple format is handy because it translates naturally into how you would graph a line on a coordinate plane.

Slope

Slope is a measure of how steep a line is. In the equation of a line, usually represented by \( m \), the slope refers to the rate at which the y-value changes as the x-value changes. Steeper lines have larger slopes. If the slope is positive, the line rises from left to right. Conversely, if the slope is negative, the line falls from left to right.You can calculate slope using the formula:\[ m = \frac{\Delta y}{\Delta x} \]What this means is that slope equals the ratio of the change in \( y \) (vertical change) to the change in \( x \) (horizontal change). If a line has a slope of 5, like in the exercise, that tells us that for every step you move to the right on the x-axis, the line moves 5 steps up the y-axis. Understanding slope is essential for interpreting how lines behave on a graph.

Y-Intercept

The y-intercept of a line is where the line crosses the y-axis. In the slope-intercept formula \( y = mx + b \), the y-intercept is represented by \( b \). It's an important value because it tells us the starting point of the line on the graph, at x = 0. In other words, if you know the y-intercept, you know where the line intersects the y-axis, which is where x is zero. In our example, the y-intercept is 2. This means that when graphed, the line will hit the y-axis at the point (0, 2). This allows you to quickly plot one point on the graph and then use the slope to determine other points the line will pass through. The y-intercept is a critical concept because it establishes the line's position relative to the y-axis, giving you a solid reference point on the graph.